Q  A 


LIBR^RV 

OK  THE 

University  of  California. 

Class 


ON  A  CERTAIN  CLASS  OF  FUNCTIONS  ANALOGOUS 
TO  THE  THETA  FUNCTIONS. 


f^^±-^%.. 


[MlTBBSITT) 


DISSERTATION 

PRESENTED  TO   THE   BOARD   OF  UNIVERSITY   STUDIES   OF  THE 

JOHNS   HOPKINS   UNIVERSITY   FOR   THE   DEGREE 

OF   DOCTOR   OF   PHILOSOPHY 

BY 

j^BRi^HAM    COHEN 

BALTIMORE 

1894 


^  B  R  A  R 
^  OF  THE 

UNIVERSITY 

OF  .     ^ 


PRESS  OF 

THE   FEIEDENWALD   COMPANY 

BALTIMORE 


.  ,8B  A  R  y 
^  OFTHt 

UNIVERSITY 


INTRODUCTION. 


M.  Appell,  in  a  brief  note  in  the  "  Annales  de  la  Faculte  des  Sciences  de 
Marseilles,"  gives  as  an  example  of  a  function  of  three  variables  having  a  true 
period  and  a  quasi-period,  analogous  to  the  (9-functions,  the  function 

where,  in  order  that  the  series  may  be  convergent,  the  real  part  of  a  is  to  be 
negative. 

This  function  evidently  satisfies  the  conditions 


<p{x-\-a,  y-{-2x-^a,  z-\-Sx -\-Sy  j-a)=e"'+^''+^'J  +  ^'^(p{x,  y,  2), 


dx      dyoz         cy        o^ 

Moreover,  as  M.  Appell  shows,  these  conditions  are  sufficient  to  determine 
<p{x,  y ,  z)  to  within  a  constant  factor. 

The  object  of  the  present  paper  is  to  investigate  the  properties  of  this 
function  and  of  functions  derived  from  it,  as  well  as  of  others  similar  to  it, 
pointing  out,  as  far  as  possible,  their  analogy  to  those  of  the  ^-functions.  As 
is  not  surprising,  some  of  the  properties  of  the  latter  seem  to  have  no  analogues 
in  the  case  of  the  functions  here  considered.  In  such  instances  it  has  been 
endeavored  to  assign  the  reason,  as  far  as  possible. 

The  great  difficulty  throughout  the  preparation  of  this  thesis  has  been  the 
utter  poverty  of  known  theorems  holding  for  functions  of  more  than  one  complex 
variable.     As  a  consequence,  this  work  has  been  rather  of  a  tentative  nature. 

In  this,  the  assistance  rendered  me  by  Professor  Craig,  at  whose  sugges- 
tion this  subject  was  selected,  was  invaluable,  my  appreciation  of  which  it  is 
only  proper  that  I  express  here.  I  desire  also  to  acknowledge  the  debt  of 
gratitude  I  owe  to  both  Professor  Craig  and  Professor  Franklin  for  their 
interest  manifested  in  my  work  throughout  my  entire  connection  with  the 
Johns  Hopkins  University. 

.}  >-^  n  o  -  ^  ■; 


I. 

hetf{xy  y,  2)  be  a  holoraorphic  function  satisfying  the  conditions 
f{x-j-(o„  y,  z)=f{x,  y-^co^,  z)=f{x,  y,  z -^  co^)  =zf  {x ,  y,  z),  (1) 

=  e  <-.*,,  0,3     /(a;^    y^    z),  \^) 


(3) 


a«      2riitOi  dydz '        dy      27ti(02  dz^ 

where  w, ,  co^ ,  wg  are  any  quantities,  real  or  imaginary, 

p  any  given  integer, 

a  a  constant  whose  real  part  is  negative. 

The  most  general  entire  function  of  a;,  y,  z  satisfying  conditions  (1)  is  given 
by  the  Fourier  series, 

f{x,y,z)=T     'I"      "f   a.z.,»e"''^<^*+4'  +  <-^'"^  (4) 


i=  — 00      /=  — 00      »=_ 


where  Ck.i.m  is  independent  of  x,  y,  z.     In  order  that /(a;,  y,  z)  also  satisfy 
conditions  (3) ,  we  must  have 

k=zlm     and     Izizm? 
or 

kzzzm^  lz=znr 

and  we  now  have  only  the  simply  infinite  series 

•    f{x,y,z)='^f  C^e'-^-r'-^.'-r'-'^r'.  (40 

Finally,  from  (2)  we  have,  on  multiplying  both  sides  of  the  equation  by 

am*  +  ap*  +  2iri  (-i  p»  +  JL  p»  +  ~p) 
6  •>!  "a  "s 

and  properly  collecting  the  terms 

^     a(m  +  p)*  +  7ni[l  (m  +  p)«  +   V.  (m +p)>  +  ±  (m  +  p)]  am* +  M(^m»+ JL  m*  +  J.  m) 


In  order  that  this  equation  be  satisfied,  it  is  evidently  necessary  and  sufficient 
that,  for  all  values  of  m, 

Hence  the  most  general  function  of  a;,  y,  z  satisfying  the  conditions  (1),  (2), 
(3)  will  be  given  by 


f{x,  y,  z)z=z    X    Ome 


0),  U]  a>3 


(6) 


with 


Oot Cm  -|-  p  ' 


Since  by  hypothesis  the  real  part  of  a  is  negative,  this  function  is  holomorphic 
for  all  values  of  a;,  y,  z. 
If  we  write 

-til  {X ,  y ,  z)  —    ^   ^  "«  "«  "«    - 


J:i'i{x,  y ,  z) —     ^6  w,  wj  w, 


Mp[Xf  y,  z) —     2^e  »>»  wu  <"3 


-tip\^>    V}    ^) Ze  "I  "a  "a 


.     (6) 


it  is  clear  that /(a;,  y,  z)  will  be  a  linear  homogeneous  function  of  i?i ,  jRg,  •  •  •  •  , 
Rp,  . . . .  Rp.  Moreover,  the  latter  are  linearly  independent,  as  may  be  seen  at 
once  from  their  development.  They  can,  however,  be  replaced  by  simpler 
functions.     Write 


./^  msj.  y 


(p{x,  y,  z)=z    z    e     ^     S     ^u,,     ^<03  '. 


(7) 


Then,  for  ^ ,  ft,  v  any  integers,  we  have 

tn  (t  -i-  ^'^'^^      11  -L  t^^'^^      ^  _L  ^"^3^ ^  am*  +  2,ri  (-?-  ms  4-  -^  w»  +  A  m)  +  ^JL*  (Xm»  +  M^'  +  vm) 

Jr  P  P  m 

or 


Giving  to  ).,  ji,  v  each  separately  all  values  from  0  to  p —  1  we  get  p^  equa- 
tions of  the  type  (8)  to  be  satisfied  by  the  p  quantities  R .  Of  these  p^  equations, 
only  p  can  be  independent.  Moreover  there  are  p  independent  ones  among 
them,  viz.  as  we  shall  see,  those  obtained  by  putting  ).=. [x=zO  and  letting  v 
take  all  integer  values  from  0  to  p  —  1 ;  these  are 

VO)  2>rty  Inipv 

<p{x,  y,  s-|---«)z=6P  B,-\-  ....  -\.e-T-Bp-\-  ....  -f  i2p 

P  (9) 

v  =  0,  1,  ....,p  —  \. 

For  the  sake  of  brevity  we  shall  introduce  the  following  notation.     Write 

<p{x,y,  z)  =  \0,0,Q)\ 
<p{^+Y,  y,  2)  =  [^,  0,  0] 

Jr 
jr 


Also  write 


[0,  0,  v]  =  C-(a5,  2/,  2)=Cv. 


(10) 


Our  equations  (9)  may  then  be  written 

[0,  0,  0]  =  Co=    i2x  +  i^2     4-----+    i2p  +  ....+i2p 

[0,  0,  i]  =  Ci  =  rii2i  +  r2^+..--  -hnRf>+-"'-\-rpRp 
[0,  0,  2]  =  C2  =  ;ii2i  +  ?1i22  +  ....+r?Rp+-.--+?i^P 

where 

3iri  4fft  2pirt 

ri  =  «*  ,  r2=ep  ,  —  ,  Yp  =  e'j' ,  —  ,  rp  =  i- 

The  independence  of  these  equations  follows  at  once  from  the  fact  that  the 
determinant  of  the  system 


J= 


1,1, 

Ti      }  Tz     , 

11)72} 


r  1       ,    /  2       ; 


rP  — 1 
/p        , 


fp 


rr^ 


n 


=  n  {xi—Ti)     *4^i 


is  evidently  different  from  zero,  the  y^B  being  the  p^^  roots  of  unity.     Of  course 

(p-l)(p-2) 

the  value  of  zP  is  (—  1)        2         ^p  . 

Denoting  the  minor  of  ^f  in  J  by  Ci_i,  t,  the  determinant  of  the  minors  is 


D  = 


^0,0      ,   M,  1      ,••••»   vp— 1, 0 

M),  1         ,    ^1,  1        ,••••,    ^p-1,  1 


Solving  equations  (10) ,  we  have 

Ji2i=  CcoCoH-  Co,  iCi  +  ^©.aCa^"  •  •  •  •  4"  ^Vp-iCp-i 

J/[2=Z  C/i,oCo  H"  Ci,  1  Ci  +  Ci,2C2  ~l~  •  •  •  •   "h  Ci,p-lCp-l 

JMp=:  Cp_i^  oCo  "h  ^-1,  iCi  ~r  Cp-i,  2  C2  "i"  ■  •  •  •  "h  ^-1,  p-i  Cp-i    J 


(11) 


The  remaining  p''  — p  functions  \_X,  //,  v]  for  ^>,  //  =  1 ,  2, ,  p  —  1  and 

v:=0,  1,  2,  . .  ..  ,  p  —  1  can  now  be  expressed  as  linear  homogeneous  func- 
tions of 

C.       .  v  =  0,  1,  2,....,p  —  l. 

From  (6)  it  is  obvious  that 


}(0  A  — 


^p(^+yS>,  z)  =  e"''TR,{x,  y,  z)=f:R.    ^ 


Rp{^  +  ^y  y,  z)z=e^'  P  Rp{x,  y,  z)  =  Rp 


(12) 


Making  these  changes  in  (11)  we  get  the  following  p  systems  of  p  —  1  equations 
each : 

n^Ri=Oo,o[^,  0,  0]+eo.:[A,  0,  1]+  ....  +(7o,p_i[;,  O,  ;>  — l]    (130 

r;jEp=c,_i,o[;.,o,o]+(7,_i,i[/(,o,  i]+....+Cp_i,p_i[;,o,2)-i](i3,) 
jR,  =  o^_,,,[?.,  0,  o]+c;,_,,,[;.,  o,  i]  +  . . .  -^o,_,,,_,ix,  o,  p-i](i3p) 

X  =  l,  2,  ....  ,p-l. 

The  determinant  of  each  of  the  systems  of  p  equations  obtained  by  taking  the 
P"  equation  of  each  of  the  above  sets  is  D,  which  is  different  from  zero  since  J 
is. 

Hence  we  can  solve  for  the  p^ — p  quantities  [?.,  0,p']foT  X=zl,  . . ..  ,  p — 1 
and  v=:  0,  1,  . . . .  ,  p —  1  in  terms  of  R^,  R^, . . . .  ,  Rp,  which,  in  turn,  can 
be  expressed  linearly  in  terms  of 

C.         v  =  0,  1,  ....p  — 1. 

Thus,  taking  the  system 

n^R^  =  Oo,o[.^,  0,  o]  +  Co,,[;.,  o,  i]+ ....  4-(7o.^_i[;.,  o,  p-i] 

f:jR,=a,,iA,  0, o]-hc,,,[;, 0,  i]+....+ci,^_,[;, 0, p-i] 


r:^ii,=  q._a.  o[^  0,  o]+G,_,. ,[/(,  0, 1]+ ... . +c;-i.  ,>-i[^  o,p-i] 

Ji^^=Q,_,,o[;,0,0],    Q,_i,i[/,0,  l]+....+C^_,,^_,[/,0,_p-l]   ) 
and  remembering  that  the  minor  of  Q^  ^.  in  D  is  ^-J^^^  Jf~^,  we  have 

And  finally,  from  (11) 

P=l      j=o 
/l=:l,  2,  ....,  p-1  v  =  0,  1,  2,  ....,  p— 1. 

In  exactly  the  same  way  we  get 

j[o,  /.,  v]= V '  k  r^rfi.-x.iZi  (15) 

/^=1,  2, ,  p  — 1         v  =  0,  1,  2, ,;>  — 1 


9 


or 


j[o,  ^,  v]  =  2rrr;q.-i.o[o,  o,  0]+ 2  rrr;q,-i.i[o,  0,  1]+.... 
+  ....+2r^V;q,-i..[o,  0,  ^]+ .... +2r?r^c;_,,^_,[o,  0,  p-i]. 

.-.   .)[;, ;.,  v]=Vr^V;c;_,,o[/i,  0,  o]  +  2r:V;q._i.i[^,  0,  i]+  ....  ' 
p=i 

H- ....  +2r^;r;q._i..[^,  0,  ^]  +  ....  +  2r^;^^c;_^^_,[/,  0,^.-1]. 

Whence,  from  (14)  we  have 

p=:l  P'  =  l  i  =  0 

+  2r^V^<^p-i.i22rA^'Vp'C'p'-i.,C, 
+  2rj:V;C;.-i.222rfr^'^p'-i.,C,- 
+ 

/,  /i=  1,  2,  ....  ,  p— 1         v  =  0,  1,  2,  ....  ,  p— 1. 
If  we  write  (14)  in  the  form 

^    P=l         j=0  ^       j=0 

the  determinant  of  any  of  the  systems  of  p  equations  obtained  by  keeping  / 
fixed  and  allowing  v  to  take  all  values  from  0  to  p  —  1  is  seen  at  once  on 
writing  it  out,  to  be 


or  since  Dzzz  Jp~^  , 


y(18  +  23+....    +    P»)-P^ 

/A  jp 


v[iP(P  +  i)]''=:  J_|   J(A)    I 


Similarly,  writing  (15)  in  the  form 

^     p  =  l  j=:0  J  =  0 

the  determinant  of  any  of  the  systems  of  p  equations  obtained  by  keeping  /i 
fixed  is  found  to  be 

(l»  +  2»+  ■  ■  .  .  +p^)  —  /  (P-^) '^^-^'  =     1     I  ^(.>x)    I 


10 


Finally,  (16)  may  be  put  in  the  form 


1    k=p—i  j=p— 1 

k=0  j=0 


(19) 


The  coefficients  here  are,  to  within  the  factor  —^,  the  elements  of  the  deter- 
minant obtained  by  taking  the  product 

We  have  thus  expressed  all  of  the  p^  quantities 

[/,  /^,  v]         /,  /^,  v  =  0,  1,  2,  ....  p  — 1. 

as  linear  homogeneous  functions  of  the  p  linearly  independent  ones 
[0,  0,  v]  =  C.         v  =  0,  1,  ....  p  —  1. 

Hence  we  see  that  every  holomorphic  function  of  a:,  y,  z  satisfying  con- 
ditions (1),  (2),  (3)  can  be  expressed  as  a  linear  homogeneous  function  of  these 
p  quantities.  From  which  follows  that  there  can  be  only  p  such  functions 
which  shall  be  linearly  independent. 

We  have  obviously 


:j{x,  y,  z)  =  <p{x,  y,  2  +  ^y') 
0(^»  y>  2  +  W3)  =  ^^(a;,  y,  z) 

CiCa;,  y,  z-^k'^-)=:._^_^{x,  y,  z) 


If  for  brevity  we  write 

ap*  -\-  2-1  [^  p3  4- 1  /  -I-  i-^]  =zE{p) 


(20) 


and  if  we  denote  the  substitution 

(x,  y,  z;  X  -\ r->,  y  -\- ^ «  -\ Ap,  z  +  -^^p  H ?o'  -\ ^-^p") 

by  Sp,  we  also  have 

S,:j{x,  y,  z)  =  e-^''"r^{x,  y,  z) 

S^:j{x,  y,  z)  =  e-^^'^-"T:j{x,  y,  z) 


11 


In  general,  if 


Xx.^.J^,  y',  ^)  =  ^(^  +  y,  2/  +  -^'  2-f -^),then 

Xx,  ^,  A^-hf'^i,  y,  z)=Xk.  ^,  X^,  y+^9.,  z)=Zx.  ^,  >>  y^  2+^*^3)=/;,.  ^.  Z-^,  y,  2) 


/'<t^i 


;^<t>2 


while  the  effect  of  the  substitution  Sp,  where  p^O  (mod  p),  is  to  change 
Xx,  ^,  vi^}  y>  2)  into  some  other  function  altogether,  in  general. 


II. 


Let  us  consider  now,  in  connection  with  the  function 


Co  =  ^(a;,  y,  z)=     X 


am*  4.  2ni  (jL  m*  +  Ji  m'  +  -L  m) 


(1) 


711  = 00 


the  functions  obtained  by  increasing  zhy  -^ ,    -~  and  by '—  respectively. 
We  may  write  these  functions  briefly 


T 


^i^,  y>  ^)  =  <po{x,  y,  z)=    2    « 


E(m) 


m=  —  X 


<p{^^  y,  ^+  t)  =  ^1  (^'  y,  ^)  =  ^  C*)™ «^^™' 


(2) 


From  what  has  preceded,  it  is  plain  that  the  functions  obtained  by  adding  to  x 
and  to  y,  respectively,  in  (1)  any  multiples  of  the  quarter  periods  corresponding 
to  them,  will  be  linear  homogeneous  functions  of  the  four  functions  (2).  In 
particular,  it  is  obvious  that 


fi^  +  '-^y  y>  ^)  =  <f{x,   2/+^S  ^)  =  <P{x,  y,  z-{-"^)=z(p^{x,  y,  z) 
and 

f  {^y  y-\-"i,  ^  +  "^)=f  («+?,  y,  2+^')=^(a^+?,  y-{-'^,  z)=9^{x,  y,  z) 


m 


or  THF. 


12 


Further,  we  have  manifestly 

^j(a5  +  fl'i,  y,  z)  =  <f^{x,  y-^fo^,  z)  =  (pj{x,  y,  z -\- (Os)=:<pj{x,  y,  z) 

S^<p,{x,  y,  z)  =  {—iye-^^^^(pf{x,  y,  z) 

8^<p,{x,  y,  z)  =  {-iy^e^(p)<p,{x,  y,  z) 
j=zO,  1,  2,  3. 
If  we  apply  the  substitution 

bi  —  {x,  y,  z,  x-t^,  y  +  ^^+  4^'  '  +  ^+4^^47rt^ 


(4) 


we  shall  get  entirely  new  functions,  for 

Si<fo{x,  y,  z)  =  e  X     e 

m  =  —  00 

>Ji<Pi\X}  y}  z)  —  e         2v  W  ^  1  "»  "s 

^1^2  (^)    2/)    ^)  —  ^  -^  I — ■"■)    ^  ">  "^^  "' 

This  suggests  the  following  functions,  which  may  be  written  briefly 


M5) 


iP,{x,  y,  z)=     S     e«""  +  *) 


)H  =:  —  CO 


Writing 

we  find  at  once 


^i(a;,  y,  2)  =  :S  (*)"*«*""*  +  *' 
^2(3;,  y,  z)  =  V  (_!)'» e&'('»  +  i) 


RjTri 


ei"*  =  />< 


^j(a;  +  wi,  y  ,  z  )  = /i(/'j ^ ^ {x ,  y,  z) 

*Pi{^  ,y  +  <'>2,z  )  =  i</>j{x,  y,  z) 

4>i{^  y  y  ,  z-\-f'>s)  =  —  </'j{x,  y,  z) 


ipi{x 


y 


,  2  +  '4')=/^^j+i(^>  y^  2) 


j=0,  1,  2,  3  and  v''4  =  ^''o 


(6) 


(7) 


«>i 


while  ^j(x-|--^,  y,  z)  and  9^^  (a; -j- — ' ,    y,  2)   are   entirely  new  functions   not 
expressible  as  linear  combinations  of  any  of  the  functions  fj  or  (pj. 


13 

It  will  be  seen  from  these  equations,  that  the  periods  of  (^'j  are  not  the  same, 
as  those  of  (p^ ;  for,  in  the  case  of  the  former, 

8wi  is  the  period  corresponding  to  a^^ 

4a^2  "    "       "  «  "  y\  (8) 

2^3  "    "       "  "  "  z) 

although  each  of  the  substitutions 

{x,  y,  z;  a;  +  4wi,  y,  z),  {x,  y,z)  x,  y-\-2co^,  z),  {x,  y,z]  x,  y,  z-\-co^)      (8') 

operating  on  (p^  has  only  the  effect  of  changing  its  sign.     The  effect  of  the  sub- 
stitution 8p  on  (pj{x,  y,  z)  is  the  same  as  that  on  ^^  (a;,  y,z),  viz  : 


But  while 
we  have 


Si<pj{cc,  y,  z)  =  {—iye-^^'^(Pj{x,  y,  z)     | 
8piP,{x,  y,  z)=:{—i)^^e-^^P)<f>j{x,  y,  z)     ] 

Si</'j{x,  y,  z)  =  {—iye-^(i^<fj{x,  y,  z) 
S2r  +  ,^j{^,.y,  z)=:(-i)i('-  +  i)6-^(^  +  i)^,(a;,  y,  z) 


(9) 


(10) 


In  general,  the  effect  of  Si  is  to  change  E{m)  into  jE'(m  -|-  1),  while  Sp  changes 
E{m)  into  E(in  -{-p).  Hence  the  effect  of  8p  on  our  functions  is  the  same  as 
that  of  8^,  to  within  an  exponential  factor  which  may  be  taken  out  from  under 
the  sign  of  summation.     Similarly  8^  changes  E{m)  into  E{m  -\-  ^),  and  S^^^i 

2 

changes  E{m)  into  E{m-^r  -\-^).  Hence  we  see  here  also  that,  to  within  an 
exponential  factor  as  above,  the  effect  of  8-^  is  the  same  as  that  of  81,  and  finally, 
that  of82rA.i  the  same  as  that  of /8'|'"+^,  or  ot 8^8^,  or  of  8^ Si- 

Changing  x  into  — x  and  z  into  — z  simultaneously  has  the  effect  of  chang- 
ing m  into  —  m  in  (p ,  and  m  into  —  m  —  1  in  </> ;  hence  it 

leaves  (p^,  (p^  and  (^'q  unaltered, 
interchanges        (p-^  and  ^g, 

changes  (pi  into  —  i(^'s , " 
^3  into  i</>i ,  and 
<p2  into  —  ^2  • 

Consequently,  as  regards  z  and  z  simultaneously,  we  see  that 

f oj  (fi}  ^di  <P\  fz,  <P\  <pz  are  even,  and 
(pi  is  odd. 


14 

Changing  the  sign  of  x  or  of  z  alone,  or  of  y  changes  the  values  of  all  the  func- 
tions in  such  a  way  that  no  conclusions  as  to  parity  can  be  drawn  in  these  cases. 
In  the  case  of  each  of  these  functions  we  may  find  those  zeros  which,  like 
the  zeros  of  the  ^-functions,  cause  the  vanishing  of  the  function  by  the  cancella- 
tion in  pairs  of  the  terms  of  the  series  defining  it.  This  we  can  accomplish  by 
the  examination  of  the  function  </>2  {x,  y,  z).     We  have,  in  fact, 

^2(0,  y,  0)  =  2  (—1)"*  e«(«  +  *)^  +  2-»  ^^  ('"  +  *)'  , 

m 

Changing  m  into  —  m  —  1  we  get 

^2(0,    y,    0)=-^{—l)-m-l  ^im+i)*  +  2.il.(m  +  i)^ 
m 
;^ ^f l)»ga(m  +  i)«+2rriiL(m  +  i)» 

Hence 

^2(0,  y,  0)  =  0. 

Or,  from  (8') ,  we  have  more  generally 

^^2(4^0;  ,   y,      Ic0s)=z0 

where  h  and  I  are  any  integers  and  y  is  anything  at  all. 

Finally,  applying  the  substitution  (Sj,  (p^i^j  2/>  z)  is  reproduced  multiplied 
by  the  finite  factor  ( — 1)'  e-^(«'  which  is  different  from  zero.     Hence  we  have 

The  most  general  set  of  zeros  of  ^o  is  then,  without  loss  of  generality,  from  (8'), 


X  z=.  4Aa>,  H i-  a 


y  =  y-\-2kco,-\-^^  V  (11) 

.  —  j.^    ,   2^32/      ,    2«;8a   3 

From  the  second  and  fifth  equations  of  (7)  and  the  second  equation  of  (10) 
we  can  write  the  following  table  of  zeros  where,  as  above, 

h,  k,  I,  q,  are  any  integers, 
y  anything  whatever. 


15 


UNIVERSt  i  .     ^ 


Zeros 
of 


2/  = 


(4/i +.2)^.^1+  ^9 


^0 


or 


. ,       ,   2awi 


3,  +  2A,„,  +  ?^=5« 


'"'=  +  2^^>  +  '^=«' 


(4A  +  l)».  +  2^"", 


y  +  2*<«,  +  ^=3' 


^1 


or 


4"^!  H — z:r^  9 


Tl% 


y  +  2^..,  +  ?^^-^9^ 


7       I   n    ^(>q      I   2ac(jo  s 


y'^2 


4Awi  A H  a 

7rt 


2/+2^c.,+-^-^;^9^ 


TTt 


;,03  +  22/^^5  +  ?^-«9^ 


(4A  +  3)..,+  2^^9 


S-"3 


or 


4Awi  -|-  ^^-  5 


71% 


3/ +  2^.., +^^9^ 


+    f^7  I        OQliOo        q 

2^W2  +  -^  9~ 


2a 


ko,-\-2y^q-\-~^"q 


Tti 


2ac 


{l  +  i)co,  +  2y'"^q-\--^q' 


m 


<fo 


J       ,   2acoj2rA-l 


,   ,       ,   Sa(Oof2r  4-1 
2/  +  ^'^'^2+-^(— f- 


m 


(,+,,..  +  2,5(2r±i)  +  2^(?r.±i; 


^1 


,        ,   2acoif2r  4-1 


,    ,        ,   daco^f 


3aw2/2r+l 


(,  +  .),,+2,5(?£+>)+?^3(2^; 


f2 


,       ,   2awW2r-\-l 


,    ,       ,  Saco2f2r-l-l 


<Ps 


,        ,   2acoJ2r-^l 


OT 


,   ,       ,    3aw2/'2r4-l 
V  +  w^UaH-  — T-^i  — J — 


^''^^  +  22/^("-f~)  +  ^-^'(^~ 


(/  +  |).3  +  2,l^f2r±i)  +  2a.3^2^ 


16 


Putting  h=zk=l=q  =  r:=0\\e  get  the  following  simple  zeros 


Zeros  of 

xz= 

y= 

2=0 

</'<> 

2fui 
or 
0 

y 
y 

0 

Wg 

2 

S^l 

or 
0 

y 
y 

0 

Wg 

4 

^3 

0 

y 

0 

</'s 

or 
0 

y 
y 

0 

3ft>3 

4 

<fo 

y  +  t7 

2  "^  47rt 

^1 

Saw2 
^   '     47:i 

(t»3  I   awg 
4  ~'"4;:t 

<fn 

a  Oil 

Tti 

,    3aci).2 

4;ri 

<Pz 

Tii 

,    3a  Wo 

y  +  4./ 

3^3   ,   aa>g 
4    "^47:* 

The  zeros  of  ^p{x,  y,  z)  might  have  been  gotten  directly,  as  follows : 

am*  +  2>rt  (—  m«  +  J^  »»»+  -  m) 


^o(«,  y,  2)  =  2« 


a (^  - TO)4  +  2.ri  [^  (^_TO)B  +  iL(^_m)»+  i-(M-m)] 


=  2e' 


The  corresponding  terms  of  these  two  series  will  be  equal  but  of  opposite 
sign  for  those  values  of  a;,  y ,  z  which  make  the  exponents  of  e  in  the  two  cases 


17 


differ  by  an  odd  multiple  of  m  for  all  values  of  m.     Such  values  of  «,  y,  z 
will  evidently  cause  ^o{^)  2/j  ^)  to  vanish.     We  are  to  have,  then 

a  (u  —  my  4-  2m  [-  (u  —  mf  -\-  ^  (u  —  mf  -^  ^{u  —  m)^ 


—  \am'  +  27ri  (5  m^  +  1  m^  +  -  m)]  =  (2)fc  +  1)  :ri, 

Oi\  (02  Ws 


which  on  reduction  becomes 


(;.  -  2m)  {  (!^  +  a/,)  (2m' -  2m;.  +  ;,  +  ;?)  +  (^;.' +  2^^  +  M^  I 

l       «*!  Wl  tt;2  Wg        J 

=  (2ifcH-l);:i. 


This  condition  is  satisfied  if 

//  is  an  odd  integer, 

Ttix  ,  hti 

^  +  "''  =  T- 


TZIX 


2my 


0) 


27iiz 


^'''  +  ^/'+^  =  C/''^.  +  2(/  +  2/'  +  l)]f 


(12) 


where  X  and  /?  are  any  integers. 

To  obtain  the  zeros  of  ^o(^)  2/>  z)  we  need  only  put 

y       anything,  as  before. 

2  wa  ^  27ri  J    '^ 


(13) 


This  can  be  readily  verified,  for,  putting  these  values  in  ^  (a;,  y,  2),  we  have 

^        e  '-V  2         TTt  /         ^  <uj        ^  V  2  a)5^27rt/      J 

TO  = CO 

a{m*  -  2raV  +  w^lS)  +  27ri  1-  w  (m  —  /oi)  +  27rt  (A  m^  +  ^  +  2p  4-  1  „  \ 
ZIZ  ^e  "a  ^2  2  / 

__g-7^v>( ;^\mg''('"-f)'-|-«("»-^')'*''»+2Ti^(»i-M)w 

because 

gjri  (Ama  +  A  +  2p  +  1)  m / -lyn 

since  for  X  even,  /m^  +  /  -|-  2,o  +  1  is  odd 

X  odd,     /m^  +  /  -|-  2^0  -j- 1  is  even  or  odd  according  as  m  is  even  or  odd. 


18 


When  m  is  replaced  hy  /j.  —  m,  the  expression  above  is  only  altered  by  having 
( — 1)"*  replaced  by  ( — 1 )''-'".     Since  /Jt  is  an  odd  integer, 

i.  e.  for  the  above  values  of  the  variables,  the  function  is  equal  to  its  negative, 
and  is  hence  equal  to  zero. 

In  the  above  it  was  stated  that  y  may  be  taken  arbitrary.  As  a  matter  of 
fact,  either  y  or  z  may  be  so  chosen,  since  these  two  variables  are,  from  (12), 
subjected  only  to  the  one  condition 


27Ti        .   27ti        /:   1   o     I  i\    •  I       J 


(14) 


If  z  be  taken  arbitrary,  our  zeros  will  be 


Ttl 


y^/A+2/>+l 


apr 


2/^ 
zz=z  anything 


2m      fjLo)^ 


(16) 


For,  on  substituting  these  values,  we  get 

am«  +  2.i  [(A  ~  ?^)m»4-  (^^t2p+ll  +  aM^__^\^,^J.„-| 
^e  L\2         nlJ       ^\        2/1  ^  27ri        m«.^  <«.      J 

m 

=  e       1°    ^e  2  2  /ncoj  fi 

The  effect  of  changing  m  into  //  —  m  is  to  replace  the  factor  e*"**"'  in  the  above 
by 

g[A  (M-m)»  +  (A+ 2p  + 1) /Li-2  (A+ 2p+ 1)  m]  Tfi 

which  may  be  written,  for  brevity,  e^"*. 
If  A  is  even,  L  is  odd 

;  is  odd,  e^""'''  =  (—1)" ,  and  e^"'  =  (— 1)*"  ~  "  • 
So  that  for  all  values  of  / 

pLiti  — gAm'rri 

Which  shows  us,  in  the  same  way  as  before,  that  (15)  is  also  a  set  of  zeros. 

The  fact  that  z  in  (13)  and  in  (11)  contained  the  arbitrary  quantity  y  might 
have  also  assured  us  that  we  could  so  choose  y  as  to  give  z  any  value  we  please, 
and  still  have  the  resulting  value  of  the  function  zero. 

The  zeros  of  ^j,  ip^y  (fz  and  those  of  (/>o,  <pi,  ^2>  <ps  can  also  be  calculated 
directly  in  the  same  way  we  have  just  found  those  of  (po{x,  y,  z),  or  they  can 


19 

be  derived  from  those  of  (p^  in  a  manner  similar  to  that  used  in  obtaining  the 
zeros  of  all  the  rest  from  those  of  (['2. 

A  comparison  of  our  set  of  zeros  for  ^0  obtained  by  the  two  methods,  which, 
in  fact  however,  are  the  same  in  principle,  will  manifestly  show  them  to  be 
identical,  if  account  be  taken  of  (3) . 

By  the  first  method,  the  simplest  zeros  were  first  obtained,  and  from  these 
we  determined  the  most  general  zeros,  by  observing  what  operations  could  be 
performed  upon  the  function  without  altering  its  value,  except,  perhaps,  as  to  a 
finite  factor  difierent  from  zero.  By  the  second  method  we  obtain  the  most 
general  set  of  zeros  at  once.     The  simplest  zeros  are  then  gotten  by  putting 

I  say  that  (13),  for  example,  is  the  most  general  set  of  zeros  possible  of  the 
kind  here  considered,  for  all  the  operations  which  leave  the  value  of  the  func- 
tion unaltered,  or  unaltered  except  as  to  a  finite  factor  other  than  zero,  are  there 
provided  for. 

Thus  /  so  enters,  that  a  change  in  it  by  an  even  integer  amount  corre- 
sponds to  a  change  in  x  and  z  by  some  multiple  of  Wj  and  Wg  respectively ; 
while  the  change  in  k  will  be  an  odd  integer  when  x  and  z  are  increased  or 

diminished  by  the  same  odd  multiple  of  -^  and  -^  respectively,  which  by  (3) 

does  not  alter  the  value  of  the  function. 

The  presence  of  ()  permits  a  change  in  z  alone  by  any  multiple  of  Wg . 

y  so  enters  in  the  value  of  z  that  if  changed  by  an  integer  multiple  of 
<t»2,  z  will  be  changed  by  an  integer  multiple  of  Wg,  and  if  altered  by  an  odd 

multiple  of  ~,  the  effect  on  z  will  be  to  change  it  by  an  odd  multiple  of-^. 

Finally,  fx  being  an  odd  integer,  it  represents  the  result  of  the  operation  of 
8p,  where  p  is  any  integer,  the  effect  of  which  is,  as  we  saw,  to  leave  the  value 
of  the  function  unaltered  except  as  to  a  finite  factor  other  than  zero. 

In  the  zeros  we  have  found,  namely  those  which  cause  the  cancellation,  in 
pairs,  of  the  terms  of  the  series,  only  y  or  z  (but  not  both  simultaneously)  may 
be  taken  arbitrary,  while  -x  cannot  be  taken  arbitrary  at  all,  since  it  enters  alone 
in  one  of  the  equations  of  condition  (12).  No  way  of  discovering  other  zeros 
has  suggested  itself;  and  it  remains  a  question  whether  other  zeros  exist  or  all 
the  zeros  are  confined  within  the  above  restrictions. 


20 


The  quotients 


^1 


fa 


III. 


S^o 


<P^ 


92  fi  ^2  ^8  ^5 

are  doubly  periodic  functions,  the  substitutions 


f2 


fa 


^2  and  {x,  y,  z)  x -\-  aco^ ,  y  -\-  fico^ ,  z  +  y^o^)  leaving  ^ 

fa 

^4    "    (ir ,  y ,  2  ;  a;  +  aio^ ,  y  +  M  >  ^  +  r^'^s)       "       —  and  ^ 

fa  fs 


unaltered. 


^1    "    {^,  y^  z)  «+8«Wi,  2/ +4/5^2,  z  4-2^^3)  " 


fo 

f2 

f2 


^4    "    {x,  y,  z)  x-\-^aco„  y  +  ^i^co,,  z-^r'^^rw,)  "       ^and%         " 

fa  9 

«,  /3,  ^  being  any  integers. 

If  now  we  turn  our  attention  to  the  derivatives  of  these  quotients  with  respect 
to  X,  y  or  2  and  inquire  whether,  analogously  to  the  elliptic  functions,  these 
derivatives  are  expressible  in  terms  of  any  combination  of  the  quotients  them- 
selves, it  would  seem  that  such  is  not  the  case. 

We  shall  first  consider  the  derivatives  with  respect  to  2 .  For  convenience 
of  reference,  the  following  tables  may  be  of  service. 

We  saw,  page  13,  that 


Consequently 


fo)  faj  i'di  fi  fs)  ^1  ^^3  are  even  as  to  x  and  2  jointly,  and  that 
yAg  is  odd. 


af 0      3f a      d4'^     d  (f  1  f s)     d  (^1  ^s)  are  ^aa   ^nd 
87'    W   Tz'    ~W^'    -^^- are  oaa,  ana 


dz 


IS  even. 


000  o 

^^<f^{x-\-io,,y,z)z=z^^iPi{x,  y-\-co^,  z)  =  ^^(pj{x,  y,  z -}- (0^)=  ^<fj{x,  y,  z) 


dz  Loz       ('H 


fj 


'  '^^  82  ~ 


-E{k) 


OZ 


Loz        (Oi  "^^  J       ^    dz  Loz       ws      J 

i  =  o,  1,  2,  3. 


21 

The  effect  of  adding  multiples  of  (o^,  w^  and  tog  to  x,  y  and  z  respectively  in 

^pis  the  same  as  given  in  (7)  in  the  case  of  (p^. 
Cz 

Let  us  now  consider 


dz  \(fj  (pi 

If  the  numerator  of  the  second  member  is  expressible  rationally  in  terms 
of  ifj  and  <f'k  U )  k=zO,  1,  2,  3)  it  will  be  a  linear  combination  of  quadratic 
functions  of  these,  for  it  has  no  infinities  for  finite  values  of  the  variables,  and 
the  effect  of  the  substitution  ;S'i  is  to  reproduce  it  multiplied  by  e~'^^'^\  More- 
over it  is.  even  as  to  x  and  z  jointly,  and  is  changed  in  sign  when  z  is  changed 
into  z-f-Ws-  Finally  it  is  reproduced  multiplied  by  the  factor  e~^^(*^  when 
operated  on  by  8^  . 

Of  all  the  36  combinations  of  <fj  and  (^'i.  taken  two  at  a  time,  only  one, 
(fo  (po  satisfies  all  these  requirements.     Besides 

is  zero  whenever  (fo  and  whenever  </>o  is.     Hence  it  would  seem  that  we  could 
write 

(T     ^'^'^  —  d'    ^^^  —  A(D     Lb 

where,  since  this  relation  must  hold  when  operated  on  by  S^  and  when  x,  y,  z 
are  altered  by  multiples  of  (o^,  co^,  (o^  respectively,  J.  is  a  constant  and  equal  to 

^,(0,  0,  0)^v'^2(0,  0,  0) 

^o(0,  0,  0)      <f>o{0,  0,  0)' 
So  that  we  have  finally 


'<p2  {x,  y,  zj 

dz[_(p-i  {x,  y,  z). 


(f,{0,  0,  0)  1^,(0,  0,  0) 

O^       :      <Po{x,y,z)  (po{x,y,z)       ,.. 

^o(0,  0,  0)      <Po{0,  0,  0)  •  (pi{x,  y,  z)  '     ^'' 


If  now  we  apply  the  substitution 

{x,  y,  z)  x'-fwi,  y,  z) 


we  shall  get 

T^ri  (  ^-^       ^2  (0>  0,  0)1-^2(0,  0,  0)  ,  ,      ,^ 

d_ [<pi{x,  y,  zn  _ ^  ^  >  'dz^'^   '  '  <fo{x,  y,  z)     </>,{x 

dzLf,  {x,  y,  z)_\       ^o(0,  0,  0)       ^o(0,  0,  0)  '  ipl{x,  y,  z) 


22 

which  must  hold  for  all  values  of  «,  y ,  z.  Substituting  the  first  set  of  zeros 
for  ^8  given  on  page  16,  this  equation  is  satisfied.  But  using  the  second  set  we 
get 

^2(0,  0,  0)^J,{0,  0,  0)      ^,(0,  0,  0)^^/.,(0,  0,  0) 
^o(0,  0,  0)       ^o(0,  0,  0)-^8(0,  0,  0)      ^Ao(0,  0,  0) 

which  is  manifestly  not  true,  for 

^i(0,  0,  0)  =  c^8(0,  0,  0) 
while 

^o(0,  0,  0)d^ip,{0,  0,  0) 

as  may  be  seen  from  the  definition  of  these  functions.     Hence  we  must  conclude 
that  a  relation  of  the  form  (1)  does  not  exist. 
Again,  if  we  consider  the  numerator  of 


dz  \(fj  (ft 

it  will  be  found  that  of  all  the  combinations  of  <pj  and  ^j  only  <po  (p2  satisfies  all 
the  conditions  that  it  does.  Moreover,  the  numerator  vanishes  for  all  our  zeros 
of  <po  and  of  (/'^ .     But  on  writing 

d_  rs^o(a;,  y,  z)~\  _ ^  ip^{x,  y,  z)     (f'^{x,  y,  z) 
dzL(p2{x,  y,  z)J  (fl{x,  2/>  2) 

where,  as  before,  B  can  only  be  a  constant,  we  shall  find,  on  substituting  the 
first  set  of  zeros  for  ^'o , 

^JO,  0,  0)^J'2{0,  0,  0) 

^  =  ^o(o,  0,  0)     </'o{o,  oro)' 

Using  the  second  set  of  zeros  for  ^o>  we  get 

fo(0,  0,  0)^s^2(0,  0,  0)^ 
^  =  ^2(0,  0,  0)      s^'olO,  0,  0) 

These  two  values  are  not  the  same,  since 

^.1(0,  0,  0)ziz<pl{0,  0,  0). 

Hence  we  conclude  that  no  relation  of  the  type  (3)  exists.  In  the  same  way, 
the  expressions  for  the  derivatives  of  all  the  various  quotients  of  a  ^'  by  a  f  in 
terms  of  the  quotients  themselves  break  down. 


23 

The  difficulty  seems  to  lie  in  the  fact  that  the  fundamental  periods  of  ^^  are 
multiples  of  those  of  ^^.  For  we  can  pass  from  (p^  to  v''j  +  i  by  either  of  the  two 
substitutions 

{x,  y,  z;  x-\r^o^,  y,  zj,     [x,  y,  z;  x,  y,  z  + —-)• 

At  the  same  time  (fj  from  which  ^^  was  derived  as  the  result  of  the  substitution 
S^,  is  left  unaltered  by  the  first  of  these  substitutions,  and  is  changed  into' 
^,+1  by  the  second. 

All  our  relations,  arrived  at  in  a  manner  similar  to  that  indicated  above, 
broke  down  when  subjected  to  this  test. 

It  should  be  noted  that  this  test  does  not  apply  in  the  case  of  the  6-  and 
^-functions.  As  no  general  theorem  analogous  to  that  made  use  of  in  this 
connection  in  the  case  of  the  ^-functions  could  be  established  for  the  functions 
here  considered,  the  above  method  was  employed,  to  show  that  no  such  relations 
exist. 

In  a  similar  way,  no  quadratic  relations  between  any  of  the  <pj  and  ^^ 
satisfying  all  the  tests  at  our  command,  could  be  found. 

The  objection,  above  mentioned,  as  holding  good  against  the  relations 
between  the  derivatives  of  the  quotients  and  the  quotients  themselves,  do  not 
seem  to  hold  in  the  following  cases,  where  only  the  quotients  of  the  (fj  or  of  the 
<p)c  are  involved  separately.     Thus*,  it  was  found  that 


8 

dz 


'(pojx,  y,  z)' 


^zi^,  y,  z):^^(po{x,  y,  z)  —  (po{x,  y,  2)^-^2(0;,  y,  z) 


and 


'<po{x,  y,  z)' 


<pl{x,  y,  2) 
^0(0,0,0)1^.(0,0,0)     ^(,/^,,)_^.(,,^,,) 
(*i{0,  0,  0)  — }«|(0,  0,  0)  •  <fi(x,y,z)  ^' 

4'2{^,  y,  z)^J'o{x,  y,  z)  —  <Po{x,  y,  z)-^^(pz{x,  y,  z) 
dz  L^2  {x,  y,  z)J  (pl{x,  y,  z)  '■ 

s^o(o,  0,  0)  1^3(0,  0,  0) 

qz (pi[x,  y,  z)  —  ^^^3(3;,  3/,  z)    ,gs 

-^1(0,  0,  0)-^U0,  0,-0)  •  ifUx,  y,z)  '^  ^ 

which  may  be  derived  from  (4)  as  the  result  of  operating  with  8^,  together  with 
the  other  relations  derived  from  them  by  all  the  operations  at  our  command, 
satisfy  all  the  tests  that  were  applied  to  them. 

These  results  are  given  here,  not  as  proved  relations,  but  as  such  which 
have  not  been  disproved. 


24 


The  derivatives  with  respect  to  x  and  y  are  more  complicated  than  those 
with  respect  to  z.  Hence  it  will  be  at  least  no  easier  to  establish  relations 
involving  the  former  than  to  establish  any  involving  the  latter.     Thus 


^^^0_2^'^_£(1)     '"^ 


92/ 


C02 
2zi 


m 


2  gE(jn+l) 


m=  —  00 


=  tlll!e-^(i)     2     (wi_i)2esw 


_g-£(l) 


.   (On  Wo 


Wo  J 


Similarly 


_  .-Eii)  \'d<fo      2^8  d(fo  ,   2;rt 
Ley       fOo   dz       (t>2 


]• 


9fo 

3a; 


^  i^  __  g_£{i)  rSfp 3  ^  3^0  I  3w8  9^0 27ri      n 

'^"  L9a;  W]  3y        wj    3z        Wi     °J 


In  some  respects  the  functions 

0O=Z<Po(p2  0,Z=Z(p,<ps  ¥o=i(f'o<p,  ?P*i=^i^8 

are  simpler  than  the  (fj  and  (/>)c.     Thus,  from  what  was  seen  before, 

00,  0^,    ¥1  are  even  as  to  x  and  2  simultaneously,  and 
¥0  is  odd. 


S,0o  =  —  e-^^(^^0, 

</>j  (a; -|- w, ,  2/  +  W2,  z  +  ws 

0j{x'{-oj^,       y,  z 

0j{     X,             y,  z  +  ws 


¥){x-\-(o^,  2/4-^2,  z  +  ws 

^^(a;  +  wi,       y,  z 

^j{     a;,         y4-"^2,  2 

r,(     a;,              y,  z  +  wj 


5-1^0  =  — «-'^*"  ^0 
S,  W,  =       e-2^(^'  '/i 

Si¥o  =  —  e-^^^^^0o 
Si¥,=       e-2^'*)(l^i 

=  ^j{^,  y,  2)  I 

—  0){x,y,z))      . 

=  —i¥j^i{x,  y,  z)  >| 
=      i¥j^i{x,  y,  z) 

.      .^=:        ^;(a;;  y,  z)  [ 


i  =  o,  1 


i  =  0,  1 


25- 


while,  as  before 


^.(^+'f,  y,  ^)aiid  r,(a.+  ^|,  y,z) 


are  entirely  new  functions,  not  expressible  in  terms  of  0^,0^,   W^,  W-^. 


9^_^    3^2  J- t.    ^^0 


fo 


8^2 27ri^ 


~] g-2£:(i) 


^2 


3fo 

La^" 


2;ri 


'«] 


er 


"iE  (1) 


>«t  +  ^^t]+S'-^^>»^^ 


a^  L  as  co^  J 

Similarly 

dz  Laz  (Os  J 

(72  L  az  wg  J 


^1 


?I^=    V2^(i)r^x_4!E! 


and 


Q  a</>o_ 

dz 

'^^  ar— 


\_dz        W3       J 


L  a^      W3    J 


g-2£(i) 


L  az     W3    J 


e-2^(i) 


_    OZ  Wg  ^J 


If  by  aid  of  these  formulae,  we  attempt  to  express  the  derivative  with  respect  to 
z  of  the  quotient  of  any  two  of  our  functions  (Pq,  </^i,  Wq,  W^  in  terms  of  any  or 
all  of  the  quotients,  we  will  meet  with  the  same  difficulties  as  before.  Thus  if 
we  take,  for  example 

dz\(Poi  'PI 

we  shall  find  that  of  all  the  combinations  of  our  functions  only  0i  W^^  behaves 
exactly  like  0^  '^ —  ^^o  ^^  when  put  to  the  test  of  all  the  above  operations, 


26 

and  besides,  the  numerator  of  our  expression  for  the  derivative  vanishes  for  all 
the  known  zeros  of  ^^  and  of  W^ .     But  on  writing 

3.  r^o(a^>  y,  2)1  _  f  ^i{x,  y,  z)      ¥,{x,  y,  z)  ,«, 

dzL0o{x,y,z)J-''  0l{x,y,z)  ^""f 

where,  as  before,  C  must  be  a  constant,  we  shall  find  that,  according  as  we  use 
the  first  set  of  zeros  or  the  second  set  of  zeros  of  ?Fo  (a; ,  y ,  2) ,  which  are  those 
o{</'o{x,  y,  2)  and  ^'2  (x,  y,  2), 

0,{o,  0,  o)|n(o,  0,  0) 


or 


0,{O,  0,  0)        r,(0,  0,  0) 
^</>,(0,  0,  0)  1 2^0  (0,0,  0) 


c= 


<i5'o(0,  0,  0)        r,(0,  0,  0) 

But  these  two  are  not  the  same,  since 

(Pl[0,  0,  Qi)z^0\{O,  0,  0). 

Hence  we  conclude  that  no  relations  of  the  type  (6)  exist. 

Quadratic  relations  between  0^,  0^,  W^y  W^  also  seem  not  to  exist  for  the 
same  reason,  viz.  because  the  periods  of  </^o  and  0^  are  smaller  than  those  of  Wq 
and  ¥^. 

It  may  be  mentioned  in  this  connection,  that  the  following  symmetrical 
quartic  relation  between  the  functions  <i^o>  ^m  ^oy  ^1  was  discovered  in  the 
course  of  the  work,  which,  as  far  as  could  be  tested,  satisfied  all  the  conditions 
imposed;  viz. 

0l0i—¥l  ¥l  —  Al0'o-\-0\-n—n'] 
when  ^  is  a  constant  whose  value  can  be  obtained  readily. 

IV. 

Consider  the  holomorphic  function 

f,{x,  y,  z)z=  n"(l-|-e2«(2fc  +  i)*  +  2-[?^  (2*4-1)''+ ^(2t+i)+*^J) 
where  the  real  part  of  a  is  negative.     It  is  obvious  that 

■1  -p  C  nil         Wj        W| 


27 
Again,  writing 

k 

we  see  at  once  that 

•      SJ,{x,  y,  z)  =  ll+e-'^-'^^^'£  +  l  +  t'U{^>  y>  ')' 

Finally,  writing 

F{x,  y,  z)=zfi{x,  y,  z).fz{x,  y,  z) 
we  have 

s^F{x, y, ^)=z^^^,,.:' .:-.::  f{^, y,  z) 

or 

8,F{x,  y,  .)  =  e-2«-2--(^+?,  +  l^'  F{x,  y,  z). 

We  also  have 

F{xV^,  y,  z)  =  F{x,  y-\-'^,  z)  =  F{x,  y,  z  +  '^)  =  F{x,  y,  z). 

If  we  put 

2a=:A,  |^=i?o   '^=ii,,    '^=Q, 

our  function  becomes 

X{x,  y,  z)  = 

n[(l+  e^  (2fc+l)3+2.f  [|-(2fc+l)^+  ^(2^+1)+  ^-^])  (1+  ^Ai2k+l)^-2.ii±  (2fc+ip-X(2fe+l)+  ^j)-| . 

Now  we  have 

X(x-j-i2i,  y,  z)=zX{x,  y-\-ili,  z)  =  X{x,  y,  z  +  £^s)  =  X{x,  y,  z) , 
and,  denoting  by  Tp  the  resulting  form  of  Sp,  viz. 

^p-{^,  y,  2,  ^-\—^^p,  y-r-jj^P-^'-^P>^-t-jj^P-t-^p-t-^^P) 

we  have 

T,Z(a.,  2/,.)  =  6-^-2'^'(^.+4  +  4  X{x,  y,  z). 

The  function  X{x,  y ,  z)  which  resembles  the  functions  already  considered, 
in  being  periodic,  and  in  being  reproduced  to  within  a  factor  on  being  subjected 
to  a  linear  substitution,  seems  to  differ  from  them  in  not  satisfying  any  simple 
differential  equation  or  equations.  There  seems  also  to  be  more  freedom  in 
obtaining  the  zeros  of  this  function.     Thus,  while  in  the  case  of  the  functions 


28 

already  considered,  only  y  or  z  separately  could  be  taken  arbitrary,  these  two 
variables  having  only  to  satisfy  one  condition,  and  x  had  to  be  chosen  subject  to 
an  independent  condition,  our  present  function  X  {x ,  y,  z)  vanishes  whenever 
either  of  the  following  conditions  is  satisfied  : 

A  {2k  +  If  +  2;ri  [^(2^  +  1)^  -f  |^ (2^  +  1)  +  ^J  =  (2/  +  1)  ;ri 
or 

A  {2k  +  If  -  2m  [-|  {2k  +  1)^  - 1  (2^  +  1)  +  j^  =  {21  +  1)  Tzi 


where  I  is  any  integer,  positive,  zero  or  negative,  and  k  is  any  positive  integer, 
including  zero.  The  analogy,  however,  between  our  functions  is  made  the 
more  striking  by  noticing  that  the  zeros  we  found  for  <fo{x ,  y ,  z)  are  also  such 
for  X{x,  y ,  z).  But  all  the  zeros  of  the  latter  are  not  included  in  the  former. 
It  may  be  interesting  to  note  still  further  the  similarities  and  the  dissimi- 
larities existing  between  our  two  classes  of  functions.  For  the  sake  of  brevity 
write 

E,  {k)  =  ^  (2^  +  If  +  27Ti  [|  (2^  +  If  +  I  (2^  +  1)  +  ^2 
^2(^)  =  ^(2^  +  l)«  +  2;ri[|(2A;  +  l)2  +  ^(2^  +  l)+^]. 
Increasing  zhy  -^  we  get  a  new  function 

And  as  before 

X{x^^,y,z)  =  X{x,y-\-^,z)  =  X{x,y,z^^-^) 

=  X{x-^^^,y^^,z  +  ^-^)=zX,{x,y,z), 

=  X(a;,  y,  z)  =  Xo(a;,  y,  z). 
The  effect  of  the  substitution  Tp  is  to  change 

E^ [k)  into  E^ {k-^p),  and  E^ {k)  into  E^ {k—p) 

—  Q-{E,{Q)+E,(l) +  ....+ E,(p-l)]X^{x,  y,  z). 


29 
And,  in  the  same  way 

The  effect  of  the  substitution 

I^-[x,y,z,x-\-    ^^.,y-t    ^^    -t    27r*''  +  ^+X^^ 

is  to  change 

El  [k]  into  El  {k  +  ^),  and  E^  {k)  into  E^  [k  —  1) 

thus  giving  rise,  as  before,  to  entirely  new  functions,  when  applied  to  Xo{x,  y,  z) 
and  Xi{x,  y,  z) : 

Tr  Xo{x,  y,  z)  =  S,{x,  y,  z)  =  ni{l  +  e^^^^  +  i))  (1  +e^>(^-J))] 

and 

T^Xiix,  y,  z)=  Si{x,  y,  2)  =  11  [(1  - e^. <*=  +  i))  (1  _e^.(«:-4))]. 

It  is  noticeable  that  instead  of  the  periods  of  S  (x ,  y,  z)  being  multiples  of 
those  of  X{x,  y,  2)  we  have 

^(3^+4-.  y,  z)=S{x,  2/  +  ^,  z)=3{x,  y,  z  +  i2,)z=E{x,  y,  z). 
Again, 

^o(a^-,  y,  z-\-~')=:Si{x,y,z), 

^li^,  y,  z-^-^)  =  Sa{x,  y,  z); 

but  there  seems  no  way  of  passing  from  S^ipCy  y,  z)  to   Si{x,  y,  z),  or  vice 
versa,  by  a  change  in  a?  or  3/ . 

As  in  the  case  o? X{x,  y,  2),  we  have 

Tp3o(x,  y,  z)=e-^^^(i)  +  ^^i^)  +  ■^■■+E^ip-i)]3o{x,  y,  z) 
Tp3i{x,  y,  2)  =  (— l)^e-[^.(i)  +  ^.(i)  +  ----+-^.(p-i)]S^(a;,  y^  2). 
Also 

^i  ^oi^,  y,  2)  =  n  [(1  +e^'('=  +  i))  (1  -f  e^»(«:-i)J] 

=  e-^^^'^X,{x,  y,  z)=TiXo{x,  y,  z) 
T^  Si{^,  y,  2)=  — e-^'WXi(a;,  y,  z)=:TiXi{x,  y,  z). 


30 

But  we  had,  by  definition 

T,Xo{x,  y,  z)z=So{x,  y,  z),     T^  X^{x,  y,  z)=E,(x,  y,  z). 

Hence  we  see  that 

TlX{x,y,z)=T,X{x,y,z) 

and  similarly 

TlS{x,y,z)=T,£(x,y,z) 

i.  e.  the  effect  of  two  successive  operations  of  T^  is  identical  with  that  of  a  single 
application  of  Tj . 

Changing  the  sign  of  x  and  z  simultaneously  interchanges  E^  and  E^. 
From  this  follows  that  X^  and  X^  are  even  as  to  x  and  z  simultaneously.  But 
for  S  {x,  y,  z),  we  have  the  values  changed,  thus 

^^{—x,  y,  —  2)  =  j-^^^^^ziy,-o(a^,  y,  z) 

77   1  \  1  — C^'<-i>;r,  , 

^x{—^y  y^  —  ^)  =  l_,g,(-i)-i(^>  2/,  2). 

In  conclusion,  it  may  be  mentioned  that  by  taking  the  logarithmic  deriva- 
tive oi  Xq{x,  y,  z)  with  respect  to  z,  we  shall  obtain  a  new  function  analogous 
to  the  Z-function  of  one  variable.     Thus,  writing 


X,{x,  y,  z)j=nU+«'^^'*+"*"^*'^4<'*+'^+2«^<'^+^>'+'^''4<'^+^'co8[27r;^  (2fc+l)2-|-^] 

1  "Cj 

we  have 

^ ^=A(x,  y,  z)  = 


-    fc=oo 
*=3  *=0 


^A  (2t +i)»+2,i^(2fc  +1)  sin  [2;r  ^  (2A;  -f  If  +  -^  ] 


This  series  is  uniformly  convergent,  since  the  real  part  of  A  is  negative, 
and  therefore  represents  a  function.     We  have,  evidently, 

A  [x  -\-Qu  y,z)  =  A  {xy  y  ■\-Qi,  z)  =  A  {x,  y,  z  +.Qi)  =  A  (x,  y,  z) 


31 


and 


a  r.-A-'. 


,_^-2.i,^  +  X+_l) 


Another  differentiation  will  give  us  a  doubly  periodic  function,  for 

|j(ar,  2/,  z)z=^^A{x-^i2„  y,  z)  =  ^^A{x,  y -j- iJ,,  z)  =  ^J{x,  y^z-i-I^,) 


and 


n^J{x,y,z)  =  ^J{x,y,z). 


The  successive  derivatives  also  have  the  same  property. 


UNIVERSITY 

i^LIFORH^> 


Biographical  Sketch. 

The  author,  Abraham  Cohen,  was  born  in  Baltimore,  Md.,  September  11, 
1870.  His  elementary  education  was  gotten  at  Scheib's  Zion  School,  where  he 
was  enrolled  from  1877  till  1883.  He  then  entered  the  Baltimore  City  College, 
and  upon  graduation,  in  1888,  was  admitted  as  a  candidate  for  the  degree  of 
Bachelor  of  Arts,  in  the  Johns  Hopkins  University.  This  degree  was  conferred 
upon  him  in  June,  1891,  and  in  the  fall  of  that  year  he  re-entered  this  Univer- 
sity, as  a  candidate  for  the  degree  of  Doctor  of  Philosophy,  selecting  Mathematics 
as  his  principal,  and  Astronomy  and  Physics  as  his  first  and  second  subordinate 
subjects  respectively.  Upon  receiving  his  Bachelor's  degree  he  was  awarded  a 
University  scholarship,  which  he  resigned  to  accept  an  appointment  as  Assistant 
in  Mathematics.  This  position  he  held  for  two  years.  During  the  past  year  he 
has  held  the  Fellowship  in  Mathematics. 


»                  14  DAY  USE 

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Rec'o  iL,i 

:> 

0Ec2  m 

1 

LD  21A-50m-4,'60                                Ti«:S^fi"lfP'/i^»i- 
(A95628l0)47GB                                  "'"''*"^,^il^'^**"'^^^ 

■fl 


w 


tMi^^S 


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'4, 


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